In addition to being essentially the only strictly proper local scoring rule [1]}, in the absence of application-specified predictive loss, \(\textrm {lpd}\)  may be seen as a “non-informative” choice [2]}, [3]}.
When \(\Pi (x^{\star }\vert \mathcal {D})\)  is assumed to be Gaussian, \(\textrm {elpd}\)  is intimately related to another proper scoring rule: the Dawid–Sebastiani score [4]}.
Namely, \(\textrm {elpd}\)  is equal to the Dawid–Sebastiani score plus a constant that does not depend on the model or the data-generating process.
Further, for Gaussian predictive distributions, the highest possible \(\textrm {elpd}\)  is obtained whenever the means and variances of \(\Pi (x^{\star }\vert \mathcal {D})\)  and \(\mathcal {P}(x^{\star })\)  are identical.
By contrast, minimizing mean square error is equivalent to only matching the means of \(\Pi (x^{\star }\vert \mathcal {D})\)  and \(\mathcal {P}(x^{\star }).\)
